PHYSICAL REVIEW E 92, 022117 (2015)

Probing local equilibrium in nonequilibrium fluids

J. J. del Pozo,* P. L. Garrido,† and P. I. Hurtado‡

Institute Carlos I for Theoretical and Computational Physics, and Departamento de Electromagnetismo y Fısica de la Materia,Universidad de Granada, 18071 Granada, Spain

(Received 14 July 2014; revised manuscript received 19 June 2015; published 10 August 2015)

We use extensive computer simulations to probe local thermodynamic equilibrium (LTE) in a quintessentialmodel fluid, the two-dimensional hard-disks system. We show that macroscopic LTE is a property muchstronger than previously anticipated, even in the presence of important finite-size effects, revealing a remarkablebulk-boundary decoupling phenomenon in fluids out of equilibrium. This allows us to measure the fluid’sequation of state in simulations far from equilibrium, with an excellent accuracy comparable to the bestequilibrium simulations. Subtle corrections to LTE are found in the fluctuations of the total energy whichstrongly point to the nonlocality of the nonequilibrium potential governing the fluid’s macroscopic behavior out ofequilibrium.

DOI: 10.1103/PhysRevE.92.022117 PACS number(s): 05.70.Ln, 47.11.−j, 05.40.−a, 65.20.−w

I. INTRODUCTION

Nonequilibrium phenomena characterize the physics ofmany natural systems, and their understanding remains a majorchallenge of modern theoretical physics. Out of equilibrium,dynamics and statistics are so intimately knotted that nogeneral bottom-up approach exists yet (similar to equilibriumensemble theory) capable of predicting nonequilibrium macro-scopic behavior in terms of microscopic physics, even in thesimplest situation of a nonequilibrium steady state (NESS)[1–7]. In contrast with equilibrium, the microscopic proba-bility measure associated to a NESS (or mNESS hereafter)is a complex object, often defined on a fractal support orstrange attractor, and is utterly sensitive to microscopic detailsas the modeling of boundary reservoirs (e.g., deterministicvs stochastic) [2,8–10]. Moreover, the connection betweenmNESSs and a nonequilibrium analog of thermodynamicpotentials is still murky at best. On the other hand, we doknow that essentially different mNESSs (resulting, e.g., fromdifferent modelings of boundary baths) describe equally wellwhat seems to be the same macroscopic NESS (or MNESSin short), defined in terms of a few macroscopically smoothfields [11]. Key for this sort of nonequilibrium ensembleequivalence (which can be formally stated via the chaotichypothesis of Gallavotti and Cohen [10]) is the notion oflocal thermodynamic equilibrium (LTE) [2,12], i.e., the factthat an interacting nonequilibrium system reaches locally anequilibrium-like state defined by, e.g., a local temperature,density, and velocity (the first two related locally via standardthermodynamics), which are roughly constant across molec-ular scales but change smoothly at much larger macroscopicscales, where their evolution is governed by hydrodynamicequations. LTE plays an important role in physics, being atthe heart of many successful theories, from classical hydrody-namics [13] or nonequilibrium thermodynamics [14] to recentmacroscopic fluctuation theory (MFT) [4–7]. The latter studiesdynamic fluctuations of macroscopic observables arbitrary

*jpozo@onsager.ugr.es†garrido@onsager.ugr.es‡phurtado@onsager.ugr.es

far from equilibrium and offers explicit predictions for thelarge deviation functions (LDFs) controlling the statistics ofthese fluctuations [15]. These LDFs are believed to play innonequilibrium a role akin to the free energy (or entropy)in equilibrium systems, establishing MFT as an alternativepathway to derive thermodynamic-like potentials out of equi-librium, bypassing the complexities associated to mNESSsand their sensitivity to microscopic details [4,5]. The LDFs soobtained exhibit the hallmarks of nonequilibrium behavior;e.g., they are typically nonlocal (or equivalently nonaddi-tive) [16] in stark contrast with equilibrium phenomenology.Such nonlocality emerges from tiny, O(N−1) corrections toLTE which spread over macroscopic regions of size O(N ),with N the number of particles in the system of interest [16].This shows that LTE is a subtle property: while correctionsto LTE vanish locally in the N → ∞ limit, they have afundamental impact on nonequilibrium LDFs in the form ofnonlocality, which in turn gives rise to the ubiquitous long-range correlations which characterize nonequilibrium fluids[4,17,18].

These fundamental insights about LTE and its role outof equilibrium are coming forth from the study of a fewoversimplified stochastic models of transport [4–7,16]. Thequestion remains, however, as to whether the emerging pictureendures in more realistic systems: Does LTE hold at themacroscopic level in fluids far from equilibrium? Can wemeasure corrections to LTE at the microscopic or fluctuatinglevel? Are these corrections the fingerprints of nonlocality?Here we answer these questions for a quintessential modelof a nonequilibrium fluid, the two-dimensional hard-diskssystem under a temperature gradient [19]. As we argue below,this model contains the essential ingredients that characterizea large class of fluids whose physics is dominated by theshort-range repulsion between neighboring particles, so weexpect some of our results to generalize also to these morerealistic model fluids. In particular, we show below thatmacroscopic LTE (MLTE) is a very strong property evenin the presence of important finite-size effects, revealing astriking decoupling between the bulk fluid, which behavesmacroscopically, and two boundary layers which sum up allsorts of artificial finite-size and boundary corrections to renor-malize the effective boundary conditions on the remaining

1539-3755/2015/92(2)/022117(11) 022117-1 ©2015 American Physical Society

J. J. DEL POZO, P. L. GARRIDO, AND P. I. HURTADO PHYSICAL REVIEW E 92, 022117 (2015)

bulk. This bulk-boundary decoupling phenomenon, togetherwith the robustness of the MLTE property, allows us to measurewith stunning accuracy the equilibrium equation of state (EoS)of hard disks in nonequilibrium simulations, even across thecontroversial hexatic and solid phases [20]. To search forcorrections to LTE, we study the moments of the velocityfield and the total energy. While the former do not exhibitcorrections to LTE, the fluctuations of the total energy do pickup these small corrections, the essential difference comingfrom the nonlocal character of the higher-order momentsof the total energy. This suggests that corrections to LTEare indeed linked to the nonlocality of the nonequilibriumfluid.

II. MODEL AND SIMULATION DETAILS

One of the simplest ways to simulate a fluid frommicroscopic dynamics consists in modeling its constituentparticles as impenetrable bodies undergoing ballistic motionbetween elastic collisions with neighboring particles. Suchan oversimplied description picks up, however, the essentialingredient underlying the physics of a large class of fluids,namely, the strong, short-distance repulsion between neighbor-ing molecules. This short-range repulsion dominates the localand global emerging structures in the fluid, as well as the natureof interparticle correlations [19,21]. In this way, hard-spheremodels and their relatives capture the physics of a largeclass of complex phenomena, ranging from phase transitionsor heat flow to glassy dynamics, jamming, or the physicsof liquid crystals and granular materials, to mention just afew [11,19–34], hence defining one of the most successful,inspiring, and prolific models of physics.

From a theoretical standpoint, hard-sphere models arealso very interesting because their low- and moderate-densitylimits are directly comparable with predictions from kinetictheory (either Boltzmann-type equations for low-density orring-kinetic theory for higher densities) [35,36]. Moreover,hard-sphere models are commonly used as a reference systemsfor perturbation approaches to the statistical mechanics ofinteracting particle systems [19,37,38]. In fact, the idea ofrepresenting a liquid by a system of hard bodies can bealready found in the work of Van der Waals: his famousequation of state was derived using essentially this principle.In addition, there exists an extensive literature on efficientevent-driven molecular dynamics algorithms for hard-spheremodels [39,40] which open the door to massive, long-timesimulations with large numbers of particles. This last featureturns out to be crucial in our case, as deviations from LTEare expected to be small and to occur at the fluctuatinglevel [4–7,16], thus requiring excellent statistics to pick upsuch a weak signal.

In this work we study a system of N ∈ [1456,8838] harddisks of radius � in a two-dimensional box of unit sizeL = 1, with stochastic thermal walls [34,41,42] at x = 0,L

at temperatures T0 ∈ [2,20] and TL = 1, respectively, andperiodic boundary conditions along the y direction. Thestochastic thermal walls work as follows: each time a particlehits one of these walls, its velocity v = (vx,vy) is randomlydrawn from a Maxwellian distribution defined by the wall

temperature T0,L:

Prob0,L(vx) = |vx |√2πT0,L

exp

(− v2

x

2T0,L

),

Prob0,L(vy) = 1√2πT0,L

exp

(− v2

y

2T0,L

),

with the additional constraint that the x component ofthe velocity changes sign. Note that we take units whereBoltzmann constant kB = 1. The above process simulates ina highly efficient manner the flow of energy in and out of thesystem from an infinite, equilibrium reservoir at the chosentemperature. On the other hand, the disks radius is defined as

� =√

η/Nπ, (1)

with η = π�2N/L2 ∈ [0.05,0.725] the global packing frac-tion. With these definitions, we can approach the N → ∞limit at fixed η and constant temperature gradient �T ≡|TL − T0|/L.

In order to characterize the inhomogeneous nonequilibriumsteady state in the fluid, we divided the system into 15 virtualcells of linear size λ = L/15 along the gradient direction andmeasured locally a number of relevant observables includingthe local average kinetic energy, virial pressure, packingfraction, etc., as well as the energy current flowing through thethermal baths and the pressure exerted on the walls. Local tem-perature is then defined via the equipartition theorem from theaverage kinetic energy per particle in each local cell. Moreover,in order to look for deviations from LTE, we also measuredmoments of the velocity field and the total energy; see Sec. IV.

In order to measure the reduced pressure QN ≡ π�2PN

(with PN the pressure), we used two different methods whichyield equivalent results (see below). On one hand, we measuredthe reduced pressure exerted by the fluid on the thermalwalls, Qw(N ), in terms of the average momentum exchangedbetween the colliding particles and the thermal wall per unitlength and unit time. On the other hand, we also measuredlocally in each cell the reduced virial pressure Qv(x; N )defined as

Qv(x; N ) = TN (x)ρN (x) + π�2

2λLτcol

⟨∑col(x)

vij · rij

⟩, (2)

with ρN (x) and TN (x) the local packing fraction and localtemperature, respectively, and where the sum is taken over allcollisions occurring during a time interval τcol in a cell centeredat x. Here vij = vi − vj is the relative velocity of the collidingpair (i,j ), rij is the vector connecting the particles centersat collision, with rij = 2�, and angular brackets represent anensemble average. In equilibrium, Eq. (2) is just the viral pres-sure [43], and our results below show that locally in a nonequi-librium steady state the above expression is a sound definitionof pressure. In fact, we will show below that both wall and virialdefinitions of pressure, Qw(N ) and Qv(x; N ), yield valuesconsistent with each other and with theoretical predictions.

Finally, note that time averages of the different observableswere performed with measurements every 10 time units for atotal time of 106–107 (our time unit was set to one collision perparticle on average), after a relaxation time of 103, which wassufficient to reach the steady state. Small corrections (∼0.1%)

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FIG. 1. (Color online) Density profiles for N = 8838, η = 0.5,and varying T0 ∈ [2,20]. Shaded areas correspond to boundary layers.Inset: Finite size effects as captured by δρN (x) ≡ ρNmax (x) − ρNmin (x),with Nmax = 8838 and Nmin = 1456, for different gradients.

due to the spatial discretization of density and temperatureprofiles are explicitly taken into account and subtracted (seeAppendix A), and statistical errors in data averages (at a 99.7%confidence level) are always plotted.

III. MACROSCOPIC LOCAL EQUILIBRIUM

Figures 1 and 2 show the density and temperature profiles,ρN (x) and TN (x), respectively, measured for N = 8838, η =0.5, and varying gradients �T . These profiles are in generalnonlinear, and similar profiles are measured for different N , η,and �T . Interestingly, temperature profiles in Fig. 2 (top) canbe fitted with high accuracy by the phenomenological law

T (x)α = ax + b (3)

with α an exponent characterizing the apparent nonlinearity.This simple law has been deduced for some two-dimensionalHamiltonian and stochastic models of heat transport [44].In our case, however, the fitted exponent α exhibits apronounced dependence on N and �T (not shown), withα ∈ [0.681,0.715] and no coherent asymptotic behavior, a traitof the strong finite-size corrections affecting the hydrodynamicprofiles. These corrections are captured, for instance, by thefinite-size excess density and temperature profiles, defined as

δfN (x) ≡ fNmax (x) − fNmin (x), (4)

with f ≡ ρ, T , and Nmax = 8838 and Nmin = 1456 the max-imum and minimum number of particles used in our simula-tions. The insets in Figs. 1 and 2 show δρN (x) and δTN (x)measured for different temperature gradients, signaling theimportance of finite-size corrections in this setting, particularlynear the boundaries and most evident for density profiles.Indeed, the thermal walls act as defect lines disrupting thestructure of the surrounding fluid, a perturbation that spreadsfor a finite penetration length toward the bulk fluid, definingtwo boundary layers where finite-size effects and boundarycorrections concentrate and become maximal; see insets inFigs. 1 and 2. Furthermore, the boundary disturbance givesrise to a thermal resistance or temperature gap betweenthe profile extrapolated to the walls, TN (x = 0,L), and the

FIG. 2. (Color online) Top: Temperature profiles for the sameconditions as in Fig. 1 and finite-size effects (inset). Bottom:Temperature gap at the hot wall vs 1/

√N for varying T0, and linear

fits to the data.

associated bath temperature T0,L. In the bottom panel of Fig. 2we study the system size dependence of this thermal gap,γ0,L(�T,N ) ≡ |T0,L − TN (x = 0,L)|, finding that

γ0,L(�T,N ) ∼ N−1/2 ∀�T. (5)

In this way, the boundary thermal gaps disappear in thethermodynamic limit when approached at constant packingfraction η and temperature gradient �T . In order to minimizeboundary corrections, we hence proceed to eliminate fromour analysis below the boundary layers by removing from theprofiles the two cells immediately adjacent to each wall, i.e.,cells i = 1,2 and 14,15 (see shaded areas in Fig. 1) [42].

As described in Sec. II, we also measured the pressure in thenonequilibrium fluid using two different methods, namely, bymonitoring the (reduced) pressure exerted by the fluid on thethermal walls, Qw(N ), and alternatively by measuring a localversion of the virial pressure for hard disks, Qv(x; N ); seeEq. (2). The latter yields pressure profiles which are constantacross the bulk of the fluid but exhibit a clear estructure atthe boundary layers; see inset in top panel of Fig. 3. This isof course expected because of the local anisotropy inducedby the nearby walls. A sound definition of the fluid’s pressureis then obtained by averaging in the bulk the virial pressureprofiles. This pressure exhibits strong finite size correctionswhich scale linearly as N−1 for each �T (see inset and bottom

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FIG. 3. (Color online) Top: Bulk-averaged virial and wall re-duced pressures as a function of �T for η = 0.5 in the N → ∞limit, and cubic fit. The light blue squared symbol (�) at �T = 0represents the equilibrium pressure predicted by the Henderson EoS;see Eq. (7). In both cases (virial and wall pressures) finite-size datascale linearly with N−1; see bottom panel. The inset shows virialpressure profiles for different N , η = 0.5 and �T = 10, which areconstant in the bulk.

panel in Fig. 3), converging to a well-defined value in theN → ∞ limit. Wall pressures similarly scale as N−1 (seebottom panel in Fig. 3), and in all cases (both for finite N

and in the N → ∞ limit) the measured values agree to a highdegree of accuracy and ∀�T with those of the virial expression(see top panel in Fig. 3). Notice that the N → ∞ data areconsistent with a cubic polynomial dependence of pressure onthe temperature gradient (see top panel in Fig. 3), and suchcubic dependence is in turn compatible with the equilibriumpressure (for �T = 0) predicted by the Henderson equationof state [19,48]; see Eq. (7) below. We hence conclude thatboth definitions of pressure are compatible with each otherand with theoretical predictions, suggesting already a sort oflocal mechanical equilibrium in the nonequilibrium fluid.

We now focus on the macroscopic notion of local thermo-dynamic equilibrium described in the introduction, an issuealready explored in early computer simulations [42]. Macro-scopic LTE implies that, locally, the density and temperaturefields should be related via the equilibrium EoS,

Q = ρT Q(ρ,T ), (6)

FIG. 4. (Color online) Top: Compressibility factor QN ≡QN/[TN (x)ρN (x)] as a function of TN (x) and ρN (x) measured forN = 2900 and varying η and T0. Bottom: QN vs ρN (x) measuredfor η = 0.5, N ∈ [1456,8838] and T0 ∈ [2,20], as well as forη ∈ [0.05,0.65], N = 2900 and T0 = 10,20, summing up a total of2530 data points. For comparison, data from previous equilibriumsimulations in literature are shown, together with the Henderson EoSapproximation (line). The inset shows a detailed comparison of arunning average of our data with equilibrium results after subtractingthe leading Henderson behavior.

where Q(ρ,T ) is the compressibility factor [19]. The top panelin Fig. 4 shows results for QN ≡ QN/[TN (x)ρN (x)] measuredout of equilibrium, as a function of ρN (x) and TN (x). Note thateach nonequilibrium simulation, for fixed (�T,η,N ), covers afraction of the EoS surface, thus improving the sampling whencompared to equilibrium simulations, which yield a singlepoint on this surface. As hard disks exhibit density-temperatureseparability (i.e., temperature scales out of all thermodynamicrelations) [19,45], the associated Q depends exclusively ondensity, meaning that a complete collapse is expected forthe projection of the EoS surface on the Q-ρ plane, as weindeed observe; see top panel in Fig. 4. Strikingly, althoughdensity and temperature profiles, as well as pressures, alldepend strongly on N (see Figs. 1–3), the measured QN as afunction of the local density exhibits no finite size correctionsat all; see bottom panel in Fig. 4, where a total of 2530data points for different N ∈ [1456,8838], η ∈ [0.05,0.65] andT0 ∈ [2,20] are shown. This strongly suggests a compellingstructural decoupling between the bulk fluid, which behavesmacroscopically and thus obeys locally the thermodynamic

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EoS, and the boundary layers near the thermal walls, whichsum up all sorts of artificial finite-size and boundary correc-tions to renormalize the effective boundary conditions on theremaining bulk. This remarkable bulk-boundary decouplingphenomenon, instrumental in the recent discovery of novelscaling laws in nonequilibrium fluids [45,46], is even moresurprising at the light of the long-range correlations present innonequilibrium fluids [4,17,18], offering a tantalizing methodto obtain macroscopic properties of nonequilibrium fluidswithout resorting to unreliable finite-size scaling extrapola-tions [46]. For comparison, we include in Fig. 4 (bottom) datafrom several extensive equilibrium simulations carried withdifferent methods during the last 60 years [20,47], as well asthe Henderson EoS approximation [19,48]

QH77(ρ) =[

1 + ρ2/8

(1 − ρ)2− 0.043

ρ4

(1 − ρ)3

], (7)

which is reasonably accurate in the fluid phase. The inset inFig. 4 shows a fine comparison of a running average of ourdata and the equilibrium simulations in literature, once theleading Henderson behavior has been subtracted. An excellentagreement is found to within 1% relative error, confirmingthe validity and robustness of macroscopic LTE and thebulk-boundary decoupling phenomenon here reported. Theaccuracy of our data for the EoS is surprising taking intoaccount that local cells have at most 500 particles, and manyfewer in the typical case.

Note that our data include points across and beyondthe controversial liquid-hexatic-solid double phase transitionregime [20]. In fact, for η � 0.6 a coexistence between a fluidphase near the hot wall and a solid-like phase near the cold oneis established. In Fig. 5 we plot a typical configuration in thisnonequilibrium coexistence regime with two color codings.The first one (top) represents the local angular order parameterψ6 [20], with

ψ6 ≡ 1

Nb

Nb∑k=1

eiφk , (8)

where Nb is the number of nearest neighbors of a given particleand φk is the angle of the bond connecting the referenceparticle with its k neighbor, relative to an arbitrary direction(x in our particular case). The order parameter ψ6 picks thelocal hexatic order of the symmetry-broken phase, offeringan interesting method to characterize the interface betweenthe inhomogeneous fluid and solid phases. This coexistenceappears in the presence of a strong temperature gradient andan associated heat current, as captured by the second colorcode, which represents kinetic energy in the bottom panel ofFig. 5. This interesting nonequilibrium fluid/solid coexistencewill be investigated in detail in a forthcoming paper [49].

IV. CORRECTIONS TO LOCAL EQUILIBRIUM AT THEFLUCTUATING LEVEL

The macroscopic notion of LTE that we have just confirmeddoes not carry over, however, to microscopic scales. Thelocal statistics associated to a fluid’s mNESS must be morecomplicated than a local Gibbs measure, containing small(but intricate) corrections which are essential for transport to

FIG. 5. (Color online) Snapshot of a typical hard-disks configu-ration with N = 7838, η = 0.7, and T0 = 10 and two color codesrepresenting respectively the local hexatic order (top) and the kineticenergy (bottom). Inhomogeneous fluid and solid phases coexist forsuch high η under a temperature gradient.

happen. In fact, the local Gibbs measure in a fluid is even invelocities, while, for instance, the local energy current is odd,thus leading to a zero average energy current for a mLTE state,in contradiction with observed behavior. It is therefore the tinycorrections to mLTE that are responsible for transport.

To search for these corrections, we first measured thelocal velocity statistics (not shown), which turns out to beindistinguishable within our precision from a local Maxwelliandistribution with the associated local hydrodynamic fields asparameters. Experience with stochastic lattice gases [12,16]suggests that deviations from LTE are more easily detected inglobal observables, so we also measured the global velocitymoments

vn ≡⟨

1

N

N∑i=1

|vi |n⟩

(9)

with n = 1,2,3,4, and compared them with the LTE predic-tions based on a local Gibbsian measure,

vlen ≡ an

η

∫ 1

0dxρ(x)T (x)n/2 (10)

(see Appendix B for a derivation), with a1 = √π/2, a2 = 2,

a3 = 3√

π/2, and a4 = 8. The LTE corrections we are lookingfor are tiny, so several correcting factors must be taken intoaccount, e.g., the effect of the discretized hydrodynamicprofiles and their fluctuations on vle

n , and the appropriateerror propagation; see Appendix B. Figure 6 shows the firstfour velocity moments as a function of �T , after subtractingthe (uncorrected) LTE contribution, as well as the N → ∞extrapolation before and after the corrections to vle

n mentioned

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-0.006

-0.004

-0.002

0

v1-v1le

-0.003

-0.001

0.001

0.003

v2-v2le

0 5 10 15 20

ΔT0

0.2

0.4

v 3-v3le

0 5 10 15 20

ΔT0

2

4

6

v 4-v4le

0 5 10 15 20 ΔT1

2

3

4v1

0 5 10 15ΔT

0

40

80

v3

0 5 10 15

ΔT

0

200

400

600

800

v4

N

N N

(a)(b)

(c) (d)

FIG. 6. (Color online) Velocity moments after subtracting the LTE contribution, as a function of �T for η = 0.5 and varying N . Filledsymbols correspond to the N → ∞ limit before (•) and after (�) correcting the LTE values for the discretization of hydrodynamic profilesand their fluctuations (see Appendix B). After the correction, no deviations from LTE are observed in velocity moments. The insets show themoments before subtracting the LTE part.

above. Although a naive comparison of velocity moments withraw LTE estimates would suggest that LTE already breaksdown at this level, it becomes apparent that, after properlyaccounting for all corrections mentioned, no deviations fromLTE are observed in velocity moments.

To further pursue the analogy with stochastic latticegases [12,16], we also studied the central moments mn(u) ≡〈(u − 〈u〉)n〉 of the fluid’s total energy per particle:

u ≡ 1

N

N∑i=1

1

2mv2

i . (11)

Figures 7(a)–7(c) show the measured mn(u), n = 1,2,3, as afunction of �T for η = 0.5 and different N , together withthe N → ∞ extrapolation of our data and the (corrected) LTEestimates for energy moments, mn(u)le (see Appendix C). Weobserve that, while the average energy does indeed follow theLTE behavior, 〈u〉 ∼ 〈u〉le, energy fluctuations [as captured bym2(u) in Fig. 7(b)] exhibit increasing deviations from the LTEestimate. In fact, the excess energy fluctuations scale linearlywith the squared gradient,

δm2(u) ≡ N [m2(u) − m2(u)le] ≈ + 1

40�T 2 (12)

[see Fig. 7(d)], a result strongly reminiscent of the behavior ob-served in schematic models like the Kipnis-Marchioro-Presutti(KMP) model of heat transport or the symmetric exclusionprocess (SEP) [16], where δm2(u) = ±�T 2/12. Interestingly,energy fluctuations for hard disks are enhanced with respect toLTE, δm2(u) > 0, as happens for the KMP model and contraryto the observation for SEP [16], although the excess amplitude

is roughly three times smaller for disks. A natural question isthen why we detect corrections to LTE in energy fluctuationsbut not in velocity moments. The answer lies in the nonlocalcharacter of energy fluctuations. In fact, while v2 includes onlya sum of local factors, m2(u) includes nonlocal contributionsof the form 〈v2

i v2j 〉, i �= j , summed over the whole system.

Small, O(N−1) corrections to LTE extending over large,O(N ) regions give rise to weak long-range correlations in thesystem which, when summed over macroscopic regions, yielda net contribution to energy fluctuations, which thus departfrom the LTE expectation [4,5,16]. In this way, the observedbreakdown of LTE at the energy fluctuation level is a reflectionof the nonlocality of the underlying large deviation functiongoverning fluctuations in the nonequilibrium fluid. This resultis, to our knowledge, the first evidence of a nonlocal LDF in arealistic model fluid, and suggests the study in more detail oflarge deviation statistics in hard disks.

A first step in this direction consists in investigating thestatistics of the energy current flowing through the systemduring a long time τ , a key observable out of equilibrium.In order to do so, and given the difficulties in samplingthe tails of a LDF, we measured the central moments ofthe time-integrated energy current for τ = 10. Figures 8(a)and 8(c) show the behavior of the first three central momentsof the current as a function of the temperature gradient,while Figs. 8(b) and 8(d) show the dependence of the firstand second current moments on the global packing fractionη. Notice that we scale the current by the radius of theparticles � in order to normalize data for different numberof particles. A first observation is that our measurements for

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0 5 10 15 20ΔT

0

2

4

6

8

u

0 5 10 15 20ΔT

0

50

100

Nm

2(u)

N=1456N=2244N=2900N=3729N=4367N=5226N=5935N=6853N=7838N=8838N=∞N=∞, LE

0 5 10 15 20

ΔT

0

2000

4000

6000

N2 m

3(u)

0 100 200 300 400

ΔT2

0

2

4

6

8

10

N[m

2(u)

-m2(

u)le]

1/40

(a) (b)

(c) (d)

FIG. 7. (Color online) (a–c) Scaled central moments of the total energy as a function of �T for η = 0.5 and varying N . Filled symbolscorrespond to the N → ∞ limit of our data (•) and after assuming LTE (�). (d) The excess energy variance (as compared to LTE) scaleslinearly with �T 2 with a slope ≈ 1/40.

the current third moment seem compatible with zero ∀�T

in the asymptotic N → ∞ limit [see Fig. 8(c)] suggestingGaussian current fluctuations within our accuracy level. Onthe other hand it is interesting to note that, while the averagecurrent (as well as the total energy moments, not shown) aresmooth functions of the global packing fraction [see Fig. 8(b)]

current fluctuations as captured by m2(j ) exhibit a remarkablestructure, with a minimum right before the liquid-to-hexaticequilibrium transition ηlh ≈ 0.7, where a first inflection pointappears, followed by another one at the hexatic-to-solidtransition ηhs ≈ 0.72; see Fig. 8(d). This shows that hints ofthe double phase transition arise in the transport properties of

FIG. 8. (Color online) (a, c, and inset) Scaled central moments of the time-averaged current vs �T for η = 0.5 and varying N . (b, d)Average current and its variance as a function of the global packing fraction for �T = 10 and N = 8838. The liquid-to-hexatic (ηlh) andhexatic-to-solid (ηhs) transition points are signaled.

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the nonequilibrium hard-disk fluid, a behavior that deservesfurther exploration [49].

V. DISCUSSION

In summary, we have probed LTE in a quintessential modelfluid, the hard-disks system, finding that macroscopic LTE isa very robust property. This is so even under strong finite-sizeeffects, due to a remarkable bulk-boundary structural decou-pling by which all sorts of finite-size and boundary correctionsare renormalized into new boundary conditions for the bulkfluid, which in turn obeys the macroscopic laws. We use theseproperties to measure with high accuracy the hard-disks EoS,even across the fluid-hexatic-solid transition regime. However,weak but clear violations of microscopic or statistical LTE arefound in the fluctuations of the total energy, which stronglysuggest that the nonequilibrium potential governing the drivenfluid’s macroscopic behavior is intrinsically nonlocal. It wouldbe therefore interesting to investigate more in depth the largedeviation statistics of hard disks, using both macroscopicfluctuation theory and simulations of rare events.

ACKNOWLEDGMENTS

P.I.H. thanks J. Hurtado Martınez for all his support. Finan-cial support from Spanish projects FIS2009-08451(MICINN)and FIS2013-43201-P (MINECO), NSF Grant DMR1104500,University of Granada, Junta de Andalucıa projects P06-FQM1505 and P09-FQM4682, and the GENIL PYR-2014-13project is acknowledged.

APPENDIX A: CORRECTIONS DUE TO DISCRETIZATIONEFFECTS IN DENSITY AND TEMPERATURE PROFILES

We measure in the steady state the local temperature (i.e.,local average kinetic energy) and local packing fraction ateach of the 15 cells in which we divide the simulation boxalong the gradient (i.e., x) direction. In order to minimizecells boundary effects, when a disk overlaps with any of theimaginary lines separating two neighboring cells, it contributesto the density and kinetic energy of each cell proportionally toits overlapping area. Note that the number of cells is constantin all simulations, independently of N , η, T0 or TL, so eachcell becomes macroscopic asymptotically as N → ∞. We nowrelate averages around a finite neighborhood of a given pointin space with the underlying continuous profiles in order tosubtract any possible bias or systematic correction from thedata. In particular, let TC and ρC be the temperature andpacking fraction in a cell centered at xc ∈ [0,L] of size �.Assuming that there exist continuous (hydrodynamic) profilesT (x) and ρ(x), we can relate the cell averages to the continuousprofiles by noting that

TC = 1

�ρC

∫ xc+�/2

xc−�/2dx ρ(x)T (x),

ρC = 1

�

∫ xc+�/2

xc−�/2dx ρ(x).

We may expand now the continuous profiles around xc insidethe cell of interest and solve the above integrals. Keeping

results up to �2 order we arrive at

TC = 1

ρC

{ρ(xc)T (xc) + �2

24

d2

dx2[ρ(x)T (x)]x=xc

+ O(�3)

},

ρC = ρ(xc) + �2

24

d2ρ(x)

dx2|x=xc

+ O(�3).

By inverting the above expressions, we obtain the desiredresult, namely,

T (xc) = TC − 1

24

[2

ρC

(ρC+1 − ρC)(TC+1 − TC)

+ TC+1 − 2TC + TC−1

], (A1)

ρ(xc) = ρC − 1

24[ρC+1 − 2ρC + ρC−1], (A2)

which yields the points of the underlying continuous profiles,T (xc) and ρ(xc), in terms of the measured observables, TC

and ρC respectively. These corrections to the cell density andtemperature are typically small (∼ 0.1%), though important todisentangle the different finite-size effects in order to obtainthe striking collapse of the hard-disks EoS described in themain text.

APPENDIX B: VELOCITY MOMENTS UNDER LOCALEQUILIBRIUM AND CORRECTIONS

The local equilibrium probability measure to observe aparticle configuration with positions ri and momenta pi ,∀i ∈ [1,N ], can be written as [12]

μle(r1, . . . ,rN ; p1, . . . ,pN )

∝ exp

⎧⎨⎩−

N∑i=1

β(εxi)

⎡⎣pi

2

2m+ 1

2

∑j �=i

�(ri − rj )

⎤⎦⎫⎬⎭, (B1)

where ε is the parameter that connects the microscopicscale with the hydrodynamic one, β(εxi) is the local inversetemperature around xi , and �(r) is the interparticle potential.Note that we have already assumed that temperature variesonly along the x direction. Under this microscopic LTEhypothesis, one may argue that the probability density forany particle to have a velocity modulus equal to v is given by

f (v) = v

η

∫ 1

0dx

ρ(x)

T (x)exp

[− v2

2T (x)

], (B2)

with η = ∫ 10 dxρ(x) the global packing fraction, ρ(x) being

its local version. Then, by definition, the local equilibriumvelocity moments follow as

vlen ≡ 〈vn〉le = an

η

∫ 1

0dxρ(x)T (x)n/2, (B3)

where a1 = (π/2)1/2, a2 = 2, a3 = 3(π/2)1/2, and a4 = 8.Now, in order to compare the measured velocity momentsin the main text with the local equilibrium expectations, vle

n ,we first have to express the latter in terms of our observables.For that, three steps should be taken, namely:

(1) Write Eq. (B3) as a function of the measured celltemperatures and densities, TC and ρC , following Eqs. (A1)–(A2).

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PROBING LOCAL EQUILIBRIUM IN NONEQUILIBRIUM . . . PHYSICAL REVIEW E 92, 022117 (2015)

(2) Take into account the fact that both TC and ρC haveerror bars and thus should be considered as fluctuatingvariables around their average. Note also that the fluctuationsof ρ(x) should be constrained to a constant average η.

(3) Consider the correct error propagation to compute theerror bars of vle

n .

1. Conversion to cell variables

We now express Eq. (B3) in terms of cell variables. In orderto proceed, we write

vlen = an

η

∑C

∫ xc+�/2

xc−�/2dxρ(x)T (x)n/2, (B4)

where xc are the center of the cells. We may expand theprevious expression up to order �2 to arrive at

vlen = an

η�∑C

(ρ(xc)T (xc)n/2 + �2

24

{d2ρ(x)

dx2

∣∣∣∣x=xc

T (xc)n/2

+ ndρ(x)

dx

∣∣∣∣x=xc

T (xc)n/2−1 dT (x)

dx

∣∣∣∣x=xc

+ n

2

(n

2− 1

)ρ(xc)T (xc)n/2−2

[dT (x)

dx

∣∣∣∣x=xc

]2

+ n

2ρ(xc)T (xc)n/2−1 d2T (x)

dx2

∣∣∣∣x=xc

}), (B5)

and using Eqs. (A1)–(A2), we obtain

vlen = �an

η

∑C

ρCTn/2C + �an

96ηn(n − 2)

×∑C

ρCTn/2−2C (TC+1 − TC)2. (B6)

2. Accounting for cell temperature and density fluctuations

Due to the finite number of measurements in simulations,any magnitude and, in particular, the cell observables willexhibit fluctuations that will affect the observed averagedbehavior. We now assume that these fluctuations are Gaussianand obey

ρC = ρC + γCξC ,

Prob(ξ1, . . . ,ξM ) =√

2πM

M∏i=1

[1√2π

e−ξ 2i /2

]δ

(M∑i=1

ξi

),

TC = TC + σCζC ,

Prob(ζ1, . . . ,ζM ) =M∏i=1

[1√2π

e−ζ 2i /2

], (B7)

where M = 15 is the number of cells, γC and σC are themeasured (empirical) errors associated to the average densityand temperature at cell C, ρC , and TC respectively, and ξC

and ζC are Gaussian random variables with zero mean andvariance one. Notice that the density noise takes into accountthe fact that the total density is constant. For the case studied

here we may assume that the errors are O(�), and we willexpand results up to orders σ 2 or γ 2. Now, by substituting ρC

and TC in Eq. (B6) by the fluctuating expressions in (B7) andthen averaging over the noise distributions, we arrive at

vlen ≡ ⟨

vlen

⟩ξ,ζ

= �an

η

∑C

ρC

⟨T

n/2C

⟩ζ+ �an

96ηn(n − 2)

×∑C

ρCTn/2−2C (TC+1 − TC)2 + O(�3), (B8)

where⟨T

n/2C

⟩ζ

= Tn/2C + 1

8n(n − 2)T n/2−2C σ 2

C + O(σ 4

C

). (B9)

It therefore becomes clear that fluctuations in each cell add asmall correction to vle

n .

3. Computing error bars in the local equilibrium approximation

Once we know how to compute vlen from our set of data, we

want to obtain their error bars, χn, defined as

χ2n = ⟨(

vlen

)2⟩ξ,ζ

− ⟨vle

n

⟩2ξ,ζ

. (B10)

In order to do so, we first need to know that

⟨ξ 2C

⟩ξ

= 1 − 1

M, 〈ξCξC ′ 〉ξ = − 1

Mif C �= C ′. (B11)

It is then easy to show that

χ2n =

(�an

η

)2[∑

C

(bnσ

2Cρ2

CT n−2C + γ 2

CT nC

)

− 1

M

(∑C

γCTn/2C

)2⎤⎦ (B12)

with b1 = 1/4, b2 = 1, b3 = 9/4, and b4 = 4. To betterappreciate the relevance of the above corrections, it is usefulto define

(vn

le)

0 ≡ �an

η

∑C

ρCTn/2C ,

(vn

le)

corr ≡ vnle − (

vnle)

0.

(B13)

That is, (vnle)corr contains the �2 effects due to the finite sizes

of the boxes and the effects of the errors on the measuredtemperature and density profiles. Though small, this correctionterm is essential to confirm that corrections to LTE does notshow up in our measurements of velocity moments; see maintext.

APPENDIX C: ENERGY CENTRAL MOMENTS UNDERTHE LOCAL EQUILIBRIUM HYPOTHESIS

To compute the local equilibrium expressions for the centralmoments of the total energy per particle let us start with theequilibrium expression for the energy fluctuations in the grand

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J. J. DEL POZO, P. L. GARRIDO, AND P. I. HURTADO PHYSICAL REVIEW E 92, 022117 (2015)

canonical ensemble

mn(U ) = (−1)n+1 ∂n

∂βnln �, (C1)

where U is the system total energy [see the Hamiltonian HN

defined in Eq. (B1)], mn(U ) ≡ 〈(U − 〈U 〉)n〉, and

� =∞∑

N=0

zNZN, ZN = 1

N !h2N

∫S

drN

∫dpNe−βHN . (C2)

For hard disks the canonical partition function ZN can bewritten

ZN = (2π )N

N !h2NβN�(N,S,r), (C3)

where � is the configurational part of the canonical partitionfunction which does not depend in this case on temperature.The energy fluctuations then follow as

m2(U ) = T 2[〈N〉 + m2(N )],(C4)

m3(U ) = T 3[2〈N〉 + 3m2(N ) + m3(N )],

with mn(N ) the central moments of the total number ofparticles in the grand canonical ensemble. Assuming now thatlocal equilibrium holds we can write

〈U 〉le =∫ 1

0dx〈Nx〉T (x),

(C5)

mn(U )le =∫ 1

0dx mn(Ux),

where Nx (Ux) is the local number of particles (local energy)at position x. In this way, if N is the total number of particlesin the system, we arrive at

ule = 〈U 〉N

= 1

η

∫ 1

0dxρ(x)T (x),

Nm2(u)le = m2(U )le

N= 1

η

∫ 1

0dxρ(x)T (x)2

[1 + m2(Nx)

〈N〉x

],

N2m3(u)le = m3(U )le

N= 1

η

∫ 1

0dxρ(x)T (x)3

[1 + 3

2

m2(Nx)

〈N〉x + 1

2

m3(Nx)

〈N〉x

]. (C6)

These expressions define the local equilibrium prediction for the first three central moments of the total energy per particle inour hard disk system. Note that the calculation of the corrections to these LTE estimates due to the discreteness of the measuredprofiles is similar to that explained in Appendix B for the velocity moments.

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